Bubble Sort Merge Sort Bubble Sort Sorting Sorting

Bubble Sort Merge Sort

Bubble Sort

Sorting • Sorting takes an unordered collection and makes it an ordered one. 1 2 3 4 5 6 77 42 35 12 101 5 1 2 3 4 5 6 5 12 35 42 77 101

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 3 4 5 6 77 42 35 12 101 5

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 3 4 5 6 42 Swap 77 42 77 35 12 101 5

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 42 2 3 35 Swap 35 77 77 4 5 6 12 101 5

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 42 35 3 4 12 Swap 12 77 77 5 6 101 5

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 3 4 5 6 42 35 12 77 101 5 No need to swap

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 3 4 42 35 12 77 5 6 5 Swap 101 5

"Bubbling Up" the Largest Element • Traverse a collection of elements – Move from the front to the end – “Bubble” the largest value to the end using pair-wise comparisons and swapping 1 2 3 4 5 6 42 35 12 77 5 101 Largest value correctly placed

The “Bubble Up” Algorithm index <- 1 last_compare_at <- n – 1 loop exitif(index > last_compare_at) if(A[index] > A[index + 1]) then Swap(A[index], A[index + 1]) endif index <- index + 1 endloop

LB No, Swap isn’t built in. Procedure Swap(a, b isoftype in/out Num) t isoftype Num t <- a a <- b b <- t endprocedure // Swap

Items of Interest • Notice that only the largest value is correctly placed • All other values are still out of order • So we need to repeat this process 1 2 3 4 5 6 42 35 12 77 5 101 Largest value correctly placed

Repeat “Bubble Up” How Many Times? • If we have N elements… • And if each time we bubble an element, we place it in its correct location… • Then we repeat the “bubble up” process N – 1 times. • This guarantees we’ll correctly place all N elements.

N-1 “Bubbling” All the Elements 1 42 2 35 3 12 4 77 5 5 6 101 1 35 2 12 3 42 4 5 5 77 6 101 1 12 3 5 3 35 4 42 5 77 6 101 1 12 2 35 2 5 4 42 5 77 6 101 1 5 2 12 3 35 4 42 5 77 6 101

Reducing the Number of Comparisons 1 77 1 42 2 35 3 12 4 77 5 101 5 5 6 101 1 35 2 12 3 42 4 5 5 77 6 101 1 12 2 35 3 5 4 42 5 77 6 101 1 12 2 5 3 35 4 42 5 77 6 101

Reducing the Number of Comparisons • On the Nth “bubble up”, we only need to do MAX-N comparisons. • For example: – This is the 4 th “bubble up” – MAX is 6 – Thus we have 2 comparisons to do 1 12 2 35 3 5 4 42 5 77 6 101

Putting It All Together

N is … // Size of Array Arr_Type definesa Array[1. . N] of Num Procedure Swap(n 1, n 2 isoftype in/out Num) temp isoftype Num temp <- n 1 <- n 2 <- temp endprocedure // Swap

Outer loop exitif(to_do = 0) index <- 1 loop exitif(index > to_do) if(A[index] > A[index + 1]) then Swap(A[index], A[index + 1]) endif index <- index + 1 endloop to_do <- to_do - 1 endloop endprocedure // Bubblesort Inner loop procedure Bubblesort(A isoftype in/out Arr_Type) to_do, index isoftype Num to_do <- N – 1

Already Sorted Collections? • What if the collection was already sorted? • What if only a few elements were out of place and after a couple of “bubble ups, ” the collection was sorted? • We want to be able to detect this and “stop early”! 1 5 2 12 3 35 4 42 5 77 6 101

Using a Boolean “Flag” • We can use a boolean variable to determine if any swapping occurred during the “bubble up. ” • If no swapping occurred, then we know that the collection is already sorted! • This boolean “flag” needs to be reset after each “bubble up. ”

did_swap isoftype Boolean did_swap <- true loop exitif ((to_do = 0) OR NOT(did_swap)) index <- 1 did_swap <- false loop exitif(index > to_do) if(A[index] > A[index + 1]) then Swap(A[index], A[index + 1]) did_swap <- true endif index <- index + 1 endloop to_do <- to_do - 1 endloop

An Animated Example N 8 to_do 7 did_swap true index 98 23 45 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 1 did_swap false 98 23 45 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 1 did_swap false Swap 98 23 45 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 1 did_swap true Swap 23 98 45 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 2 did_swap 23 98 45 14 1 2 3 4 true 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 2 did_swap true Swap 23 98 45 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 2 did_swap true Swap 23 45 98 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 3 did_swap 23 45 98 14 1 2 3 4 true 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 3 did_swap true Swap 23 45 98 14 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 3 did_swap true Swap 23 45 14 98 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 4 did_swap 23 45 14 98 1 2 3 4 true 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 4 did_swap true Swap 23 45 14 98 1 2 3 4 6 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 4 did_swap true Swap 23 45 14 6 1 2 3 4 98 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 5 did_swap 23 45 14 6 1 2 3 4 true 98 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 5 did_swap true Swap 23 45 14 6 1 2 3 4 98 67 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 5 did_swap true Swap 23 45 14 6 1 2 3 4 67 98 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 6 did_swap 23 45 14 6 1 2 3 4 true 67 98 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 6 did_swap true Swap 23 45 14 6 1 2 3 4 67 98 33 42 5 6 7 8

An Animated Example N 8 to_do 7 index 6 did_swap true Swap 23 45 14 6 1 2 3 4 67 33 98 42 5 6 7 8

An Animated Example N 8 to_do 7 index 7 did_swap 23 45 14 6 1 2 3 4 true 67 33 98 42 5 6 7 8

An Animated Example N 8 to_do 7 index 7 did_swap true Swap 23 45 14 6 1 2 3 4 67 33 98 42 5 6 7 8

An Animated Example N 8 to_do 7 index 7 did_swap true Swap 23 45 14 6 1 2 3 4 67 33 42 98 5 6 7 8

After First Pass of Outer Loop N 8 to_do 7 index 8 did_swap Finished first “Bubble Up” 23 45 14 6 1 true 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 1 did_swap false 23 45 14 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 1 did_swap false No Swap 23 45 14 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 2 did_swap false 23 45 14 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 2 did_swap false Swap 23 45 14 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 2 did_swap true Swap 23 14 45 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 3 did_swap 23 14 45 6 1 2 3 4 true 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 3 did_swap true Swap 23 14 45 6 1 2 3 4 67 33 42 98 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 3 did_swap true Swap 23 14 6 1 2 3 45 67 33 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 4 did_swap 23 14 6 1 2 3 true 45 67 33 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 4 did_swap true No Swap 23 14 6 1 2 3 45 67 33 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 5 did_swap 23 14 6 1 2 3 true 45 67 33 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 5 did_swap true Swap 23 14 6 1 2 3 45 67 33 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 5 did_swap true Swap 23 14 6 1 2 3 45 33 67 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 6 did_swap 23 14 6 1 2 3 true 45 33 67 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 6 did_swap true Swap 23 14 6 1 2 3 45 33 67 42 98 4 5 6 7 8

The Second “Bubble Up” N 8 to_do 6 index 6 did_swap true Swap 23 14 6 1 2 3 45 33 42 67 98 4 5 6 7 8

After Second Pass of Outer Loop N 8 to_do 6 index 7 did_swap Finished second “Bubble Up” 23 14 6 1 true 2 3 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 1 did_swap false 23 14 6 1 2 3 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 1 did_swap false Swap 23 14 6 1 2 3 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 1 did_swap true Swap 14 23 6 1 2 3 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 2 did_swap 14 23 6 1 2 3 true 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 2 did_swap true Swap 14 23 6 1 2 3 45 33 42 67 98 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 2 did_swap true Swap 14 6 1 2 23 45 33 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 3 did_swap 14 6 1 2 true 23 45 33 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 3 did_swap true No Swap 14 6 1 2 23 45 33 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 4 did_swap 14 6 1 2 true 23 45 33 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 4 did_swap true Swap 14 6 1 2 23 45 33 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 4 did_swap true Swap 14 6 1 2 23 33 45 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 5 did_swap 14 6 1 2 true 23 33 45 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 5 did_swap true Swap 14 6 1 2 23 33 45 42 67 98 3 4 5 6 7 8

The Third “Bubble Up” N 8 to_do 5 index 5 did_swap true Swap 14 6 1 2 23 33 42 45 67 98 3 4 5 6 7 8

After Third Pass of Outer Loop N 8 to_do 5 index 6 did_swap Finished third “Bubble Up” 14 6 1 true 2 23 33 42 45 67 98 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 1 did_swap false 14 6 1 2 23 33 42 45 67 98 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 1 did_swap false Swap 14 6 1 2 23 33 42 45 67 98 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 1 did_swap true Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 2 6 1 did_swap true 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 2 did_swap true No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 3 6 1 did_swap true 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 3 did_swap true No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 4 6 1 did_swap true 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fourth “Bubble Up” N 8 to_do 4 index 4 did_swap true No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

After Fourth Pass of Outer Loop N 8 to_do 4 index 5 6 1 did_swap true Finished fourth “Bubble Up” 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 1 6 1 did_swap false 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 1 did_swap false No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 2 6 1 did_swap false 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 2 did_swap false No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 3 6 1 did_swap false 14 23 33 42 45 67 98 2 3 4 5 6 7 8

The Fifth “Bubble Up” N 8 to_do 3 index 3 did_swap false No Swap 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

After Fifth Pass of Outer Loop N 8 to_do 3 index 4 6 1 did_swap false Finished fifth “Bubble Up” 14 23 33 42 45 67 98 2 3 4 5 6 7 8

Finished “Early” N 8 to_do 3 index 4 did_swap false We didn’t do any swapping, so all of the other elements must be correctly placed. We can “skip” the last two passes of the outer loop. 6 1 14 23 33 42 45 67 98 2 3 4 5 6 7 8

Summary • “Bubble Up” algorithm will move largest value to its correct location (to the right) • Repeat “Bubble Up” until all elements are correctly placed: – Maximum of N-1 times – Can finish early if no swapping occurs • We reduce the number of elements we compare each time one is correctly placed

LB Truth in CS Act • NOBODY EVER USES BUBBLE SORT • NOBODY • NOT EVER • BECAUSE IT IS EXTREMELY INEFFICIENT

Questions?

Mergesort

Sorting • Sorting takes an unordered collection and makes it an ordered one. 1 2 3 4 5 6 77 42 35 12 101 5 1 2 3 4 5 6 5 12 35 42 77 101

Divide and Conquer • Divide and Conquer cuts the problem in half each time, but uses the result of both halves: – cut the problem in half until the problem is trivial – solve for both halves – combine the solutions

Mergesort • A divide-and-conquer algorithm: • Divide the unsorted array into 2 halves until the sub-arrays only contain one element • Merge the sub-problem solutions together: – Compare the sub-array’s first elements – Remove the smallest element and put it into the result array – Continue the process until all elements have been put into the result array 37 23 6 89 15 12 2 19

How to Remember Merge Sort? That’s easy. Just remember Mariah Carey. As a singing star, Ms. Carey has perfected the “wax-on” wave motion--a clockwise sweep of her hand used to emphasize lyrics. The Maria “Wax-on” Angle: ¦(q, t) The Siren of Subquadratic Sorts

How To Remember Merge Sort? ¦(q, t) Just as Mariah recursively moves her hands into smaller circles, so too does merge sort recursively split an array into smaller segments. We need two such recursions, one for each half of the split array.

Algorithm Mergesort(Passed an array) if array size > 1 Divide array in half Call Mergesort on first half. Call Mergesort on second half. Merge two halves. Merge(Passed two arrays) Compare leading element in each array Select lower and place in new array. (If one input array is empty then place remainder of other array in output array)

LB More TRUTH in CS • We don’t really pass in two arrays! • We pass in one array with indicator variables which tell us where one set of data starts and finishes and where the other set of data starts and finishes. • Honest. s 1 f 1 s 2 f 2

LB Algorithm Mergesort(Passed an array) if array size > 1 Divide array in half Call Mergesort on first half. Call Mergesort on second half. Merge two halves. Merge(Passed two arrays) Compare leading element in each array Select lower and place in new array. (If one input array is empty then place remainder of other array in output array)

98 23 45 14 6 67 33 42

98 23 45 14 6 67 33 42

98 23 45 14 6 67 33 42

98 23 45 14 6 67 33 42

98 23 45 14 98 23 Merge 45 14 6 67 33 42

98 23 45 14 98 23 23 Merge 45 14 6 67 33 42

98 23 45 14 98 23 23 98 Merge 45 14 6 67 33 42

98 23 45 14 98 23 23 98 45 14 6 67 33 42

98 23 45 14 23 98 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 Merge 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42

98 23 45 14 98 23 23 98 45 14 6 67 33 42 6 67 14 45 14 23 45 98 Merge 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 67 6 Merge 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 67 Merge 33 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 42 6 67 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 Merge

98 23 45 14 98 23 23 98 45 14 14 45 6 67 33 42 6 33 67 6 67 42 33 42 14 23 45 98 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 6 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 6 33 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 6 33 42 Merge

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 67 Merge

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 6 67 33 42 45 14 45 23 98 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 33 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 33 42 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 33 42 45 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 33 42 45 67 Merge 42

98 23 45 14 6 67 33 42 98 23 45 14 98 23 23 98 6 67 33 42 45 14 6 67 33 42 6 33 14 45 67 6 67 33 42 67 14 23 45 98 6 14 23 33 42 45 67 98 Merge 42

98 23 45 14 98 23 23 98 45 14 14 45 14 23 45 98 6 67 33 42 6 33 67 6 67 42 33 42 67 6 14 23 33 42 45 67 98

98 23 45 14 6 67 33 42 6 14 23 33 42 45 67 98

Summary • Divide the unsorted collection into two • Until the sub-arrays only contain one element • Then merge the sub-problem solutions together

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